It's actually not due to our Neanderthal DNA (though, it did help adapt to the northern climate). It was actually the discovery of agriculture that did it, about 10,000 years ago https://www.discovermagazine.com/the-sciences/they-dont-make-homo-sapiens-like-they-used-to. Genetic markers indicate that the rate of brain evolution increased to about 100x faster than any time in human history. Ironically, our last common ancestor with Africans was ~50,000 years ago, and white skin evolved about 10,000 as well.